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Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

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Claim. The implication of Problem 1167, in the site's wording with no condition on γ\gamma or on the κα\kappa_\alpha, is false. Take γ=1\gamma=1 and κ0=λ+\kappa_0=\lambda^+. With a single color, the premise 2λ→(λ+)1r+12^\lambda\to(\lambda^+)^{r+1}_1 asks only for a subset of 2λ2^\lambda of size λ+\lambda^+, which exists since 2λ>λ2^\lambda>\lambda. The conclusion λ→(λ+)1r\lambda\to(\lambda^+)^r_1 asks for a subset of λ\lambda of size λ+\lambda^+, which does not exist. Zeraoulia's note, A note on Erdős-Hajnal-Rado Problem #1167: a counterexample to the literal statement and remarks on the intended formulation, self-published on ResearchGate under the DOI linked above, gives this counterexample and discusses the conditions under which the question is meant.

Covers. The site's wording only; the corrected Statement of the problem page is not touched. That Statement adds the conditions 2≤r<ω2\le r<\omega, γ≥2\gamma\ge2 and κα>r\kappa_\alpha>r of the Erdős–Hajnal list, and it is open.

Depends on. No page of this wiki.

Why it is rejected. It answers the site's wording, not the corrected Statement. Problem 1167 judges its corrected Statement, which adds the conditions γ≥2\gamma\ge2 and κα>r\kappa_\alpha>r; the problem page's Notes give the evidence for that reading. The counterexample has γ=1\gamma=1, so it settles no instance of the corrected Statement. The problem page's Notes credit the result.

Independent check. The formal-conjectures statement file for the problem proves the same counterexample (γ=1\gamma=1, κ0=ℵ1\kappa_0=\aleph_1, λ=ℵ0\lambda=\aleph_0) as its test lemma erdos_1167.unrestricted_is_false. That file is a statement file, not a formalization of this note, and the corpus has not built it, so it gives no formalized evidence.

Acceptance. None documented. The note has no journal record. The site labels the problem OPEN, and its curator's reply in the discussion thread points to the conditions of Komjáth's Problem 2 rather than accepting a disproof, so the reply is not acceptance. The claim is rejected, and acceptance would not change that, since the rejection concerns what the result answers.